Key primary_f

uel

Hg_Min_lb_

per_MWh_

Rank

Reason_for_Removal ORIS code Plant Name

52071_5B coal 1 52071 Sandow Station

6180_1 coal 2 6180 Oak Grove

52071_5A coal 3 52071 Sandow Station

55076_AA-001 coal 4 55076 Red Hills Generating Facility

55076_AA-002 coal 5 55076 Red Hills Generating Facility

7030_U2 coal 6 7030 Twin Oaks Power One

6147_2 coal 7 6147 Monticello

6030_1 coal 8 6030 Coal Creek

6183_SM-1 coal 9 6183 San Miguel

2817_1 coal 10 2817 Leland Olds

6469_B1 coal 11 6469 Antelope Valley Station









physical_

state

NonContI

ndicator

Unit 

Number

boiler_i

d

Unit Type boiler_type MWe_cap

acity

TX 5B 5B Fluidized bed firing Fluidized bed firing 282.4

TX OG1 1 Conventional Boiler Tangential firing 817.0

TX 5A 5A Fluidized bed firing Fluidized bed firing 282.4

MS 001 AA-001 Fluidized bed firing Fluidized bed firing 250.0

MS 002 AA-002 Fluidized bed firing Fluidized bed firing 250.0

TX U2 U2 Fluidized bed firing Fluidized bed firing 172.0

TX 2 2 Conventional Boiler Tangential firing 583.0

ND Unit 1b 1 Conventional Boiler Tangential firing 619.0

TX SMEC1 SM-1 Conventional Boiler Wall firing - opposed firing 440.0

ND 1 1 Conventional Boiler Wall firing - opposed firing 230.0

ND AVS Unit 1 B1 Conventional Boiler Tangential firing 450.0









max_heat_i

nput

Unit_heat_

rate

Fuel_types fuel Fuel_Conc_l

b/MMBtu

Mercury - 

ERT

Mercury 

MW - ERT

Mercury - 

ETTS

Mercury 

MW - 

ETTS

 2,960.0 10.5 lignite lignite 3.1E-05 5.6E-06 5.9E-05

 8,970.0 11.0 lignite lignite 2.1E-05 1.1E-06 1.2E-05

 2,960.0 10.5 lignite lignite 3.0E-05 3.7E-06 3.9E-05

 2,401.0 9.6 lignite lignite 2.4E-05 1.1E-05 1.0E-04

 2,382.3 9.5 lignite lignite 2.4E-05 1.1E-05 1.0E-04

 1,717.0 10.0 lignite lignite 2.8E-05 6.9E-06 6.9E-05

 7,000.0 12.0 lignite lignite 6.7E-06 8.0E-05

 6,015.0 9.7 coal - lignite lignite

 4,980.0 11.3 coal - lignite lignite

 2,622.0 11.4 coal - lignite lignite

 6,275.0 13.9 coal - lignite lignite









Mercury - 

CEMS

Mercury - 

Part II

Mercury 

MW - Part 

II

Mercury - 

min

Mercury 

MW - min

control_group_1 control_type_1

1.0E-06 1.1E-05 1.0E-06 1.1E-05 NOx control Selective Noncatalytic Reduction

3.9E-06 4.3E-05 1.1E-06 1.22E-05 NOx control Selective Catalytic Reduction

1.4E-06 1.5E-05 1.4E-06 1.52E-05 NOx control Selective Noncatalytic Reduction

3.9E-06 3.7E-05 3.9E-06 3.72E-05 PM control Fabric Filter, unspecified

3.9E-06 3.7E-05 3.9E-06 3.72E-05 PM control Fabric Filter, unspecified

6.9E-06 6.86E-05 PM control Fabric Filter, reverse air

6.7E-06 8.04E-05 NOx control Selective Noncatalytic Reduction

1.1E-05 1.1E-04 1.1E-05 0.000106 PM control

Electrostatic precipitator, cold 

side, w/o flue gas conditioning

1.1E-05 1.2E-04 1.1E-05 0.000123 PM control

Electrostatic precipitator, cold 

side, w/o flue gas conditioning

5.3E-03 6.0E-02 5.3E-03 0.06 PM control

Electrostatic precipitator, cold 

side, w/o flue gas conditioning

5.1E-03 7.1E-02 5.1E-03 0.071 SO2 control Dry FGD - Spray Dryer









install_date_1 control_group_2 control_type_2 install_date_

2

control_group_3

01-Jul-09 Other control Activated carbon injection 01-Aug-09 SO2 control

01-Aug-09 Other control Activated carbon injection 01-Aug-09 PM control

01-Aug-09 Other control Activated carbon injection 01-Aug-09 SO2 control

01-May-02

01-May-02

01-Aug-90

01-Nov-08 Other control Activated carbon injection 01-Feb-09 PM control

01-Aug-79 SO2 control Wet FGD - Spray Type 01-Aug-79

01-Jan-82 SO2 control Wet FGD - Tray Type 01-Jan-82

01-Nov-74

01-Jul-84 PM control Fabric Filter, reverse air 01-Jul-84









control_type_3 install_date_

3

control_group_4 control_type_4 install_date_4

Dry FGD - Circulating Dry Scrubber01-Jul-09 PM control Fabric Filter, pulse 01-Jul-09

Fabric Filter, pulse 01-Aug-09 SO2 control Wet FGD - Spray Type 01-Aug-09

Dry FGD - Circulating Dry Scrubber 01-Aug-09 PM control Fabric Filter, pulse 01-Aug-09

Fabric Filter, shake and deflate 01-Dec-75 PM control Electrostatic precipitator, cold side, w/ flue gas conditioning 01-Dec-75









control_grou

p_5

control_type_5 install_date_5 control_group_6 control_type_6 install_date_6









control_group_7 control_type_7 install_date_7 control_group_8 control_type_8 install_date_8









control_g

roup_9

control_t

ype_9

install_d

ate_9

control_gr

oup_10

control_t

ype_10

install_d

ate_10

control_

group_1

1

control_t

ype_11

install_d

ate_11

control_

group_1

2









control_t

ype_12

install_d

ate_12









facility_name

facility_

id

boiler_i

d

configuration_i

d compound Final_EF

Final_EF_

Units

Final_Outp

ut_EF

Sandow Station 52071 5B 5B Mercury 1.0E-06 lb/MMBtu 1.1E-05

Sandow Station 52071 5B 5B Mercury 1.2E-06 lb/MMBtu 1.3E-05

Sandow Station 52071 5B 5B Mercury

Invalid Test 

Avg lb/MMBtuInvalid Test Avg

Oak Grove 6180 1 OG1 Mercury 1.1E-06 lb/MMBtu 1.2E-05

Oak Grove 6180 1 OG1 Mercury 3.9E-06 lb/MMBtu 4.3E-05

Sandow Station 52071 5A 5A Mercury 1.4E-06 lb/MMBtu 1.5E-05

Sandow Station 52071 5A 5A Mercury 1.9E-06 lb/MMBtu 2.0E-05

Sandow Station 52071 5A 5A Mercury 3.7E-06 lb/MMBtu 3.9E-05

Red Hills Generating Facility 55076 AA-001 001 Mercury 3.9E-06 lb/MMBtu 3.7E-05

Red Hills Generating Facility 55076 AA-001 001 Mercury 1.1E-05 lb/MMBtu 1.0E-04

Red Hills Generating Facility 55076 AA-002 002 Mercury 3.9E-06 lb/MMBtu 3.7E-05

Red Hills Generating Facility 55076 AA-002 002 Mercury 1.1E-05 lb/MMBtu 1.0E-04

Twin Oaks Power One7030 U2 U2 Mercury 6.9E-06 lb/MMBtu 6.9E-05

Monticello 6147 2 2 Mercury 6.7E-06 lb/MMBtu 8.0E-05

Coal Creek 6030 1 Unit 1b Mercury 1.1E-05 lb/MMBtu 1.1E-04

San Miguel 6183 SM-1 SMEC1 Mercury 1.1E-05 lb/MMBtu 1.2E-04

San Miguel 6183 SM-1 SMEC1 Mercury 1.9E-05 lb/MMBtu 2.2E-04

Leland Olds 2817 1 1 Mercury 5.3E-03 lb/MMBtu 6.0E-02

Antelope Valley Station6469 B1 AVS Unit 1 Mercury 5.1E-03 lb/MMBtu 7.1E-02














Final_Output_

EF_Units From Report_id

Hg_Min_lb_per_

MWh_Rank New Souce 

MACT Floor 

Test_ID

New Source MACT Floor 

QA Calc (lb/MWh)

lb/MW Part II 2-Source Emissions Survey, Stacks 1 2_Hg_T2 1.1E-05

lb/MW Part II 3-Source Emissions Survey, Inlet Ducts and Stacks 1 2_Hg_T3 1.3E-05

lb/MW ETTS 1

Invalid Test 

Avg 5.8E-05 (One Valid Run)

lb/MW ETTS 2

lb/MW Part II 1-Compliance Test Report 2

lb/MW Part II 2-Source Emissions Survey, Stacks 3

lb/MW Part II 3-Source Emissions Survey, Inlet Ducts and Stacks 3

lb/MW ETTS 3

lb/MW Part II 06-05 4

lb/MW ERT 4

lb/MW Part II 06-05 5

lb/MW ERT 5

lb/MW ERT 6

lb/MW ETTS 7

lb/MW Part II DryFining Baseline\01 8

lb/MW Part II Demonstration of Sorbent Enhancement 9

lb/MW Part II Assessment of Mercury Control Options 9

lb/MW Part II LOS Mercury Alstom Demo 10

lb/MW Part II AVS Mercury Demo 11














New Source Floor 

Agreement of QA 

Calc

Run 1 

(lb/MWh)

Run 2 

(lb/MWh)

Run 3 

(lb/MWh)

100.00% 7.0E-06 9.9E-06 1.5E-05

100.00% 9.8E-06 1.3E-05 1.6E-05

100.30% 5.8E-05 Failed Spike Recovery Failed Spike Recovery Run 1 is a Far Outlier based on Tukey's tests for near and far out














key's tests for near and far outliers.
Runs

52071

Sandow

5B

1 7.05E-06

2 9.85E-06

3 1.53E-05

4 9.84E-06

5 1.32E-05

6 1.59E-05

7 5.84E-05 Outlier

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Emissions

0.00E+00

1.00E-05

2.00E-05

3.00E-05

4.00E-05

5.00E-05

6.00E-05

7.00E-05

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 7.00

Kurtosis 6.30

SE Kurtosis 1.00

Result Kurtosis non normal

Skewness 2.47

SE Skewness 0.79 RESULT

Result Skewness non normal non normal

### Make sure that kurtosis and skewness are copied in "Calculations for Template", in cells H14 and H12 respectively
ni = number test runs =

70000000000

Total Number of sources =1

n =Total # test runs =76

This step is for QC only:

means 1.85E-05

1.85E-05

Mean =



2

2

1

1

1

n

i

i

s xX

n



   

1 1

1

i

N

n

ij

i j

X x n    

1 1

n m

2 1 1

s

n m

    

 

1

1

N

i

i

dfn



 

      

2 1 1

s

n m

    

 

1

1

n

i

i ik

i

k

x x n   

1N

i

i

n

N

Pooled Variance = 3.20E-10

m= number future runs =3

Term1 0.48

Term2 1.52E-10

Squared root Term2 1.23E-05  2 2 1 1 1 n i i

s xX

n



   

1 1

1

i

N

n

ij

i j

X x n    

1 1

n m

2 1 1

s

n m

    

 

1

1

N

i

i

dfn



 

      

2 1 1

s

n m

    

 

1

1

n

i

i ik

i

k

x x n   

1N

i

i

n

N
NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 3.14

7

0.38

99.13%

Conclusion

interval of differences for t-values used 0.453

5.86

Adjusted UPL for skewness and kurtosis for all sample size 9.08E-05 lb/MWh

9.08E-02 lb/GWhr

Confidence Level is not 99%, Go to recalculate tab to fix the Confidence Level

new t-statistic from tab Recalculate t-statistic

Adjustment for non-normality (skewness and kurtosis) for any sample size in particular for small samples 

a) n = number of test-runs =

b) Calculate u0

Current Confidence Level is

2

1

0

1

1

u

t

n







df t

2 1 1

s

n m

    

 

2

,

1 1 X tdf p

s

n m

 

     

9.08E-02 lb/GWhr

df t

2 1 1

s

n m

    

 

2

,

1 1 X tdf p

s

n m

 

     
(When Outlier is included, data distribution is non-normal)

0 1 2 3 4 5 6 7 8

1

2

3

4

5

6

7

4 and H12 respectively
Part II Data Part II Data Part II Data Part II Data Part II Data

Comments

Use Run in UPL 

Calc? Grade Key

Facility_Nam

e

BDL Yes A 2_Hg_T2_R1

BDL Yes A 2_Hg_T2_R2

BDL Yes B 2_Hg_T2_R3

Yes A 2_Hg_T3_R1

Yes A 2_Hg_T3_R2

Yes A 2_Hg_T3_R3

Part III Part III Part III Part III Part III

Used only a 3-point traverse. Ask EPA to determine if Run 1 should be included in UPL run analysis (yields non-normal data diYes C 2_Hg_T1_R1

Used only a 3-point traverse. Failed spike recovery. Do not use in any UPL analyses. No Exclude 2_Hg_T1_R2

Used only a 3-point traverse. Failed spike recovery. Do not use in any UPL analyses. No Exclude 2_Hg_T1_R3

RTI

Conversions

1 ppm SO2= 1.66E-07 lb/scf

1 ppm Hg= 5.20E-07 lb/scf

1000 mg= 1 g

1000000 ug= 1 g

454 g= 1 lb

1 m^3= 35.3145 ft^3

200.59 Hg molecular weight

64.06 SO2 molecular weight

Table 19-2 Fd Fw Fc

Fuel Type dscf/MMBtu wscf/MMBtuscf/MMBtu

Coal

Anthracite 10100 10540 1970

Bituminous 9780 10640 1800

Lignite 9860 11950 1910

Oil 9190 10320 1420

Subbituminous 9820 1840

Petroleum Coke 9830 1850

QA Calcs

Column 1 Column 2 Column 3 Column 4 Column 5

Column Source==> 43 43 43

Test Avgs for UPL R1 R2 R3

Test_IDs lb/MW lb/MW lb/MW lb/MW

2_Hg_T2 1.1E-05 7.0E-06 9.9E-06 1.5E-05

2_Hg_T3 1.3E-05 9.8E-06 1.3E-05 1.6E-05

2_Hg_T1 Invalid Test Avg 5.8E-05 Invalid Run Invalid Run
Anthracite Anthracite

Bituminous Bituminous

Lignite Lignite

Oil Oil

Pet Coke Pet Coke

Subbituminous Sub Bituminous

Part II Data Part II Data Part II Data Part II Data Part II Data Part II DataPart II Data

Facility_ID Boiler_ID

Fuel1 

Used Fuel1 Fraction

Fuel2 

Used

Fuel2 

Fractio

n

"Fuel 

Used/Fraction" 

Data Source

52071 Lignite 100% Part II Submittal

52071 Lignite 100% Part II Submittal

52071 Lignite 100% Part II Submittal

52071 Lignite 100% Part II Submittal

52071 Lignite 100% Part II Submittal

52071 Lignite 100% Part II Submittal

Part III Part III Part III Part III Part III Part III Part III

52071 Lignite ETTS

52071 Lignite ETTS

52071 Lignite ETTS

As Submitted



Column 6 Column 7 Column 8 Column 9 ####### ##### Column 12

43 43 43 43 43 43

R4 R5 R6 R7 R8 R9 R10

lb/MW lb/MW lb/MW lb/MW lb/MW lb/MW
fPM 30B μg/dscm No μg/dscm 2.205E-09

Se M5 lb/dscf Yes ppm 2.205E-12

Pb M5B ppm

Hg M29 ppm @ 3% O2

HCl OTM27 ppm @ 7% O2

SO2 26A

M6C

Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data

Pollutant

Reference 

Method

Concentrati

on

Concentr

ation 

UOM

NonDetect? MDLMDL 

UOM

HAP 

Conversion 

Value to lbs

Hg M29 Yes 0.879 μg/dscm 2.20E-09

Hg M29 Yes 1.253 μg/dscm 2.20E-09

Hg M29 Yes 2.033 μg/dscm 2.20E-09

Hg 30B 1.238 μg/dscm No 2.20E-09

Hg 30B 1.657 μg/dscm No 2.20E-09

Hg 30B 2.018 μg/dscm No 2.20E-09

Part III Part III Part III Part III Part III Part III Part III Part III

Hg 30B 7.62 μg/dscm No 2.20E-09

Hg 30B 8.63 μg/dscm No 2.20E-09

Hg 30B 6.69 μg/dscm No 2.20E-09



Column 13 Column 14 Column 15 ####### ######## ####### ######## Column 20

Test_IDs

2_Hg_T2

2_Hg_T3

2_Hg_T1
lbs/μg 0.0283 cm/cf

lb/mg

lb/scf

Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data

HAP 

Conversion 

UOM

Volume 

Conversion 

Value to 

cubic feet Reserve Reserve Reserve

Volume 

Conversion 

UOM

Bws 

(percent) %O2d

lbs/μg 0.0283 cm/cf 17.46 4.6

lbs/μg 0.0283 cm/cf 17.53 4.6

lbs/μg 0.0283 cm/cf 17.85 4.8

lbs/μg 0.0283 cm/cf 18.13 4.2

lbs/μg 0.0283 cm/cf 4.55

lbs/μg 0.0283 cm/cf 4.55

Part III Part III Part III Part III Part III Part III Part III Part III

lbs/μg 0.0283 cm/cf 19.67 3.27

lbs/μg 0.0283 cm/cf 20.47 3.31

lbs/μg 0.0283 cm/cf 19.76 3.39

RTI-QA



Column 21 Column 22 Column 23 Column 24 Column 25 Column 26 Column 27 ########
lb/hr lb/MMBtu

lb/MWh

Part II Data Part II DataPart II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data

%CO2d

Emissio

n Rate 

Value

Emission 

Rate UOM

Emission 

Rate MDL

Emission 

Factor 

Value

Emission 

Factor 

UOM

Emission 

Factor MDL Fd Fw Fc

15.2 lb/hr 0.0021 7.05E-06 lb/MWh 9860 11950 1910

15.6 lb/hr 0.003 9.85E-06 lb/MWh 9860 11950 1910

15.3 lb/hr 0.0048 1.53E-05 lb/MWh 9860 11950 1910

15.6 3.0E-03 lb/hr 9.84E-06 lb/MWh 9860 11950 1910

15.425 4.0E-03 lb/hr 1.32E-05 lb/MWh 9860 11950 1910

15.425 5.0E-03 lb/hr 1.59E-05 lb/MWh 9860 11950 1910

Part III Part III Part III Part III Part III Part III Part III Part III Part III Part III

16.22 5.57E-06 lb/MMBtu 9860 11950 1910

15.94 6.31E-06 lb/MMBtu 9860 11950 1910

16.08 4.92E-06 lb/MMBtu 9860 11950 1910

As Submitted



####### ###### ######## ######## ######## ######## Column 35 ######## ####### #####
From 

Min 

Test

Sheet

Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data Part II Data

Final for UPL

Calculated 

conc 

lb/dscf

19-1

lb/MW

%diff

19-1

19-6

lb/MW

%diff

19-6

lb/MMBtu 

Value Used for 

UPL Calcs

Unit Heat 

Rate 

(MMBtu/MW

h)

5.48E-11 7.26E-06 3.0% 7.22E-06 2.4% 6.93E-07 10.5

7.82E-11 1.04E-05 4.9% 1.00E-05 1.8% 9.88E-07 10.5

1.27E-10 1.70E-05 10.2% 1.66E-05 7.9% 1.62E-06 10.5

7.72E-11 9.99E-06 1.5% 9.91E-06 0.8% 9.53E-07 10.5

1.03E-10 1.37E-05 3.0% 1.34E-05 1.3% 1.30E-06 10.5

1.26E-10 1.66E-05 4.6% 1.63E-05 2.9% 1.59E-06 10.5

Part III Part III Part III Part III Part III Part III Part III

lb/MMBtu lb/MMBtu lb/MMBtu

4.75E-10 5.56E-06 -0.26% 5.60E-06 0.48% 5.57E-06 10.5

5.38E-10 6.31E-06 -0.06% 6.45E-06 2.22% 6.31E-06 10.5

4.17E-10 4.91E-06 -0.19% 4.96E-06 0.74% 4.92E-06 10.5

RTI-QA



######## Column 40 Column 41 Column 42 Column 43 Column 44 Column 45
original database export===>

original database export===>

original database export===>

original database export===>

original database export===>

original database export===>

original database export===>

original database export===>

Final for UPL original database export===>

lb/MWh Value 

Used in UPL Calcs

original database export===>

Key

RTI 

Calculated 

QA

lb/MMBtu Run Grade

Submitted 

Value

7.05E-06 2_Hg_T2_R1 6.89E-07 A 0.0021

9.85E-06 2_Hg_T2_R2 9.57E-07 A 0.003

1.53E-05 2_Hg_T2_R3 1.58E-06 B 0.0048

9.84E-06 2_Hg_T3_R1 9.53E-07 A 0.003

1.32E-05 2_Hg_T3_R2 1.28E-06 A 0.004

1.59E-05 2_Hg_T3_R3 1.55E-06 A 0.005

Part III Part III Part III Part III Part III Part III

RTI 

Calculated 

QA

lb/MMBtu Run Grade

Submitted 

Value

lb/MWh Value 

Used in UPL Calcs Key

5.8E-05 100.3% 2_Hg_T1_R1 5.56E-06 A 5.57E-06

6.6E-05 original database export===> 2_Hg_T1_R2 6.31E-06 Exclude 6.31E-06

5.2E-05 original database export===> 2_Hg_T1_R3 4.91E-06 Exclude 4.92E-06

original database export===>

original database export===>



Column 46 Column 47
Submitted 

Value 

UOM Use ? Comment facility_id

configurat

ion_id report_id

sampling_

port_id order

pollutant_

id

pound per hour (lb/hr) Yes BDL 52071 5B 2-Source Emissions Survey, Stacks SP14 12 408130

pound per hour (lb/hr) Yes BDL 52071 5B 2-Source Emissions Survey, Stacks SP14 12 408130

pound per hour (lb/hr) Yes BDL 52071 5B 2-Source Emissions Survey, Stacks SP14 12 408130

pound per hour (lb/hr) Yes 52071 5B 3-Source Emissions Survey, Inlet Ducts and Stacks SP14 12 408130

pound per hour (lb/hr) Yes 52071 5B 3-Source Emissions Survey, Inlet Ducts and Stacks SP14 12 408130

pound per hour (lb/hr) Yes 52071 5B 3-Source Emissions Survey, Inlet Ducts and Stacks SP14 12 408130

Part III Part III Part III Part III Part III Part III Part III Part III Part III

Submitted 

Value 

UOM Use ? Comment facility_id

configurat

ion_id

submittal_

id Hg_type

run_numb

er start_date

lb/MMBtu Yes Used only a 3-point traverse. 52071 5B 832 30B 1 40246.72

lb/MMBtu No Used only a 3-point traverse. Failed spike recovery. 52071 5B 832 30B 2 40247.55

lb/MMBtu No Used only a 3-point traverse. Failed spike recovery. 52071 5B 832 30B 3 40248.51




7.05E-06

9.85E-06

1.53E-05

pollutant_

name run_id

start_datetim

e end_datetime unit_load

fuel_flow_

rate

fuel_heati

ng_value

fg_moistu

re_conten

t

fg_oxygen

_content

mercury 10 02-Nov-09 02-Nov-09 298.08 341180 7830 17.46 4.6

mercury 2 31-Oct-09 01-Nov-09 304.55 357154 8232 17.53 4.6

mercury 6 01-Nov-09 01-Nov-09 314.07 356244 7905 17.85 4.8

mercury 19 11/1/2009 11/1/2009 305 18.13 4.2

mercury 20 11/1/2009 11/1/2009 302

mercury 21 11/2/2009 11/2/2009 315

Part III Part III Part III Part III Part III Part III Part III Part III Part III

end_date

measured

_value_Hg

measured_val

ue_Hg0

measured_val

ue_Hg+2

measured

_value_Hg

P basis

test_meth

od data_flag

method_c

omments

μg/dscm μg/dscm μg/dscm μg/dscm

40246.8 7.62 Dry U.S. EPA Method 30B ADL Based on values in previo

40247.63 8.63 Dry U.S. EPA Method 30B ADL

40248.59 6.69 Dry U.S. EPA Method 30B ADL




fg_CO2_c

ontent

fg_temper

ature

test_meth

od conc

conc_unit

s conc_ND conc_MDL conc2

conc2_uni

ts

15.2 217 EPA Methods 1, 2, 3A, 4, 5B, 6C, 7E, 9, 10, 18, 25A, 26A, 29, 201A, CTM-027, and OTM-037 microgram per dry standard cubic meter (μg/dscm) TRUE 0.879

15.6 213 EPA Methods 1, 2, 3A, 4, 5B, 6C, 7E, 9, 10, 18, 25A, 26A, 29, 201A, CTM-027, and OTM-029 microgram per dry standard cubic meter (μg/dscm) TRUE 1.253

15.3 215 EPA Methods 1, 2, 3A, 4, 5B, 6C, 7E, 9, 10, 18, 25A, 26A, 29, 201A, CTM-027, and OTM-033 microgram per dry standard cubic meter (μg/dscm) TRUE 2.033

15.6 224 EPA Methods 1, 2, 3A, 4, 6C, 17, 26A, 30B 1.238 microgram per dry standard cubic meter (μg/dscm) FALSE

EPA Methods 1, 2, 3A, 4, 6C, 17, 26A, 30B 1.657 microgram per dry standard cubic meter (μg/dscm) FALSE

EPA Methods 1, 2, 3A, 4, 6C, 17, 26A, 30B 2.018 microgram per dry standard cubic meter (μg/dscm) FALSE

Part III Part III Part III Part III Part III Part III Part III Part III Part III

emission_

rate

steam_ou

tput

heat_inpu

t

fg_moistu

re_conten

t

fg_CO2_c

ontent 19-6

%Diff

19-6

Ffactor 

calculated 19-9

lb/MMBtu

5.57E-06 1977 2484 19.67 16.22 5.6E-06 -0.00479 4.5E-06

6.31E-06 1993 2487 20.47 15.94 6.45E-06 -0.02216 5.13E-06

4.92E-06 1990 2480 19.76 16.08 4.96E-06 -0.00739 3.98E-06




emission_

rate

emission_

rate_units

emission_

rate_ND

emission_

rate_MDL

Calculated 

lb/MWh 

at Prom

Boiler 

Heat Rate 

(lb/MMBt

u)/MWh

Calculated 

lb/MMBtu 

at Prom

Calculated 

conc 

lb/dscf

19-1

lb/MMBtu

pound per hour (lb/hr) TRUE 0.0021

pound per hour (lb/hr) TRUE 0.003

pound per hour (lb/hr) TRUE 0.0048

0.003 pound per hour (lb/hr) FALSE

0.004 pound per hour (lb/hr) FALSE

0.005 pound per hour (lb/hr) FALSE

Part III Part III Part III Part III Part III Part III Part III Part III Part III

%Diff

19-9

fg_oxygen

_content 19-1

%Diff

19-1

Ffactor 

calculated 19-5

%Diff

19-5

fg_flow_r

ate

fg_temper

ature

0.192856 3.27 5.56E-06 0.002617 4.46E-06 0.198803 579345 192

0.187079 3.31 6.31E-06 0.000622 5.02E-06 0.205194 563279 194

0.191667 3.39 4.91E-06 0.001865 3.94E-06 0.199097 580823 193




if use Run 

1 O2% for 

all runs

% Diff

19-1

19-6

lb/MMBtu

if use Run 

1 O2% for 

all runs

% Diff

19-6

emission_

factor

emission_

factor_uni

ts

emission_

factor_ND

emission_

factor_M

DL

FALSE

FALSE

FALSE

FALSE

FALSE

FALSE

Part III Part III Part III Part III Part III Part III Part III Part III Part III

fuel1_inp

ut_rate

fuel1_typ

e

fuel1_rate

_units

fuel1_met

hod

fuel2_inp

ut_rate

fuel2_typ

e

fuel2_rate

_units

fuel2_met

hod

fuel3_inp

ut_rate

5544 lignite ton/day gas fuel meter

5616 lignite ton/day gas fuel meter

5592 lignite ton/day gas fuel meter




run_pollut

ant_com

ment Fuel1 Fuel1%Btu Fuel2 Fuel2%Btu

QA 

Comment

Part III Part III Part III Part III Part III Part III Part III Part III Part III

fuel3_typ

e

fuel3_rate

_units

fuel3_met

hod

fuel4_inp

ut_rate

fuel4_typ

e

fuel4_rate

_units

fuel4_met

hod

other_info

rmation

exclude_d

ata

FALSE

FALSE

FALSE




Part III Part III Part III Part III Part III

Paired 

Trap RD % 

(<10%)

Spike 

Recovery 

% (bet. 85 

and 115%)

QA 

Comment

Questions 

for EPA

0.797937 108 Used only a 3-point traverse.

6.427881 168 Used only a 3-point traverse. Failed spike recovery. Exclude this run/test? This is not Sandow's lowest test, so it will remain the number one un

3.847204 136 Used only a 3-point traverse. Failed spike recovery. Exclude this run/test? This is not Sandow's lowest test, so it will remain the number one un




average of 

test runs

1.07E-05

1.30E-05

o it will remain the number one unit.

o it will remain the number one unit.
Hg

Runs

52071

Sandow

5B

1 7.05E-06 2 9.84E-06 3 9.85E-06 4 1.32E-05 5 1.53E-05 6 1.59E-05

Outlier Removed (This 

was a single run test 

because 2 of 3 runs 

failed spike 

recovery).==> 89

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 6.00

Kurtosis -1.69

SE Kurtosis 1.03

Result Kurtosis normal

Skewness -0.14

SE Skewness 0.85

Result Skewness normal

ni = number test Runs =60000000000

Total Number of sources =1

n =Total # test Runs =6

Emissions (lb/MW)

Use for large sample sets and small sample sets that are Normal. 1N i i n  N

090000648116ceac.xlsx UPL Pooled Template_noOutlier 11/19/2012
5

This step is for QC only:

means 1.2E-05

1.2E-05

Mean =

Pooled Variance = 1.2E-11

m= number future Runs = 3

Term1 0.50

Term2 6.1E-12

Squared root Term2 2.5E-06

NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 3.36

UPL POOLED VARIANCE = 2.02E-05 lb/MWh

2.02E-02 lb/GWh

 

2

2

1

1

1

n

i

i

s x X

n 

 

 

1 1

1

i N n

ij

i j

X x

n  

 

1 1

n m



2 1 1

s

n m

 

    

1

1

N

i

i

df n



 

      

df t

2 1 1

s

n m

 

    

1

1

ni

i ik

i k

x x

n 

 

1

i

i

n





2

,

1 1 X tdf p

s

n m

 

     

090000648116ceac.xlsx UPL Pooled Template_noOutlier 11/19/2012
Runs Parameters for Tukey's Test==> 25th Percentile 75th Percentile IQ Dist. 1.5 x IQD 3 x IQD

Subcategory 2_Hg 52071

Sandow

5B

Outliers 

Removed 9.8E-06 1.6E-05 5.7E-06 8.6E-06 1.7E-05

2_Hg_T2_R1 7.05E-06

2_Hg_T3_R1 9.84E-06

2_Hg_T2_R2 9.85E-06

2_Hg_T3_R2 1.32E-05

2_Hg_T2_R3 1.53E-05

2_Hg_T3_R3 1.59E-05

2_Hg_T1_R1 5.8E-05 When the outlier was removed the curve became normal and lognormal. Far Outlier High (By Tukey's Test)

8 With the outlier in the set. The data are neither normal nor lognormal. 

9 This was a single run test because 2 of 3 runs failed spike recovery

10 (See "Lignite_NonNormal_UPL" for the nonNormal UPL Analysis")

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data Log Data

Sample Size 6.00 6.00

Kurtosis -1.69 -0.88

SE Kurtosis 1.03 1.03

Result Kurtosis normal normal

Skewness -0.14 -0.51

SE Skewness 0.85 0.85

Result Skewness normal normal

Runs

1 -11.9

2 -11.5

3 -11.5

4 -11.2

5 -11.1

6 -11.1

7

8

9

Emissions (lb/MW)

Use for small sample sets that are Lognormal.

1. Take LN( ) of Raw Data

y = 5E-06ln(x) + 6E-06

R2 = 0.909

0.00E+00

2.00E-06

4.00E-06

6.00E-06

8.00E-06

1.00E-05

1.20E-05

1.40E-05

1.60E-05

1.80E-05

0 1 2 3 4 5 6 7

Series1

Log. (Series1)

0.00E+00

1.00E-05

2.00E-05

3.00E-05

4.00E-05

5.00E-05

6.00E-05

7.00E-05

0 1 2 3 4 5 6 7

Series1

Series2

Outlier Removed

090000648116ceac.xlsx Lignite_Logormal UPL_noOutlier 11/19/2012

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

6

Average of Raw Data 1.19E-05

3. Calculate -11.38

2. Calculate sample size, n=

1

ˆ

n

i

i

x

n

 



 

2

2 1

ˆ

1

n

i

i

x x

n













090000648116ceac.xlsx Lignite_Logormal UPL_noOutlier 11/19/2012
9.85E-02

4. Calculate

5. m 3 = number of future runs to be used to assess compliance with the standard

6. Calculate

4.82

3.22E-02

151.9

7. Calculate

0.58

9. substitue the values of in cell D7 and F7 respectively

93 is the Row number in Column AE with the first value > 0.99.

3.48 z value from "lognormal distribution 99% "

a- Calculate 1.20E-05

b- Calculate 1.43E-10

c- Calculate 1.04E-01

8. Go to tab distribution for 99 percentile "lognormal distribution 99% "

10. In column AE indentify the value that is the smallest value that is larger than 0.99, note what row number this is.

11. In column B, go down to the row number from Step 10. above, and copy the z value.

12. calculate the UPL using the formula 

b- denominator first term

a- Numerator first term

 

2

2 1

ˆ

1

n

i

i

x x

n













222 4 3 ˆˆˆ 2

e e2 3e 3

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2 z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )





2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

2

4 3 ˆˆˆ 2

2 2

ˆ

233 1

3 1

1

z

e e e

m m e







  

    

  

090000648116ceac.xlsx Lignite_Logormal UPL_noOutlier 11/19/2012

c- Calculate 1.04E-01

d-calculate 1.74E-02

e- Calculate 8.19E-06

f- Calculate UPL 2.15E-05 lb/MW 2.15E-02 lb/GW

 

2

2 1

ˆ

1

n

i

i

x x

n













222 4 3 ˆˆˆ 2

e e2 3e 3

 

 

2

2

ˆ m e 1





2z

 

2 2

ˆ ˆ

1

1 2

z

e e

m

 



 



2 z  1z 

2

ˆ 2 2 2

2

2 4

ˆ

.99 2 2 ˆ ˆ ˆ ˆ 2 ˆ

ˆ ˆ ( 1)

2( 1)

z

UPL e me e m e

m n n

       

   

      

  

2

ˆ

2

ˆ

e

 

2

2 ˆ ˆ

e

 

2

ˆ

( 1 e )





2 4

ˆ ˆ

n n 2( 1)

      

  

2 2 2

2 4

2 2 ˆ ˆ ˆ ˆ 2 ˆ ˆ ˆ ( 1)

2( 1)

me e m e

n n

         

    

  

 

2 2 2

2

4 3 ˆˆˆ 2

2 2

ˆ

233 1

3 1

1

z

e e e

m m e







  

    

  

090000648116ceac.xlsx Lignite_Logormal UPL_noOutlier 11/19/2012
∆z z

0.101 151.9333 0.576654

-5 =f(a) 2956.683

1 -4.899 2690.581

2 -4.798 2441.923

3 -4.697 2209.966

4 -4.596 1993.982

5 -4.495 1793.258

6 -4.394 1607.098

7 -4.293 1434.82

8 -4.192 1275.759

9 -4.091 1129.264

10 -3.99 994.6991

11 -3.889 871.4463

12 -3.788 758.9013

13 -3.687 656.4756

14 -3.586 563.5963

15 -3.485 479.706

16 -3.384 404.2628

17 -3.283 336.7403

18 -3.182 276.6275

19 -3.081 223.4291

20 -2.98 176.665

21 -2.879 135.871

22 -2.778 100.598

23 -2.677 70.41261

24 -2.576 44.89694

25 -2.475 23.64854

26 -2.374 6.280479

27 -2.273 -7.57869

28 -2.172 -18.2849

29 -2.071 -26.1786

30 -1.97 -31.5847

31 -1.869 -34.8127

32 -1.768 -36.1565

33 -1.667 -35.8945

34 -1.566 -34.2898

35 -1.465 -31.5897

36 -1.364 -28.0262

37 -1.263 -23.8158

 

 2  1 3

336

1 3

6 24

z z

z z

 

    

 

  

 

2z

1z
38 -1.162 -19.1595

39 -1.061 -14.2426

40 -0.96 -9.23514

41 -0.859 -4.29162

42 -0.758 0.44903

43 -0.657 4.863335

44 -0.556 8.843327

45 -0.455 12.29654

46 -0.354 15.14599

47 -0.253 17.33021

48 -0.152 18.80323

49 -0.051 19.53455

50 0.05 19.50922

51 0.151 18.72773

52 0.252 17.2061

53 0.353 14.97586

54 0.454 12.08401

55 0.555 8.593056

56 0.656 4.581013

57 0.757 0.141383

58 0.858 -4.61683

59 0.959 -9.56912

60 1.06 -14.5755

61 1.161 -19.4804

62 1.262 -24.113

63 1.363 -28.2866

64 1.464 -31.7994

65 1.565 -34.4337

66 1.666 -35.9567

67 1.767 -36.1198

68 1.868 -34.659

69 1.969 -31.2949

70 2.07 -25.7325

71 2.171 -17.6613

72 2.272 -6.75539

73 2.373 7.326806

74 2.474 24.94222

75 2.575 46.46332

76 2.676 72.27807

77 2.777 102.7899

78 2.878 138.4179

79 2.979 179.5964

80 3.08 226.7754

81 3.181 280.4204

82 3.282 341.0124

83 3.383 409.0477

84 3.484 485.0383
85 3.585 569.5117

86 3.686 663.0109

87 3.787 766.0943

88 3.888 879.3358

89 3.989 1003.325

90 4.09 1138.666

91 4.191 1285.981

92 4.292 1445.904

93 4.393 1619.088

94 4.494 1806.199

95 4.595 2007.92

96 4.696 2224.948

97 4.797 2457.998

98 4.898 2707.797

99 4.999 2975.091
distribution of Z

Normal 

distribution

1.48672E-06 0.004395759

2.45094E-06 0.006594452

3.9995E-06 0.009766478

6.46025E-06 0.014276926

1.03291E-05 0.020596014

1.63472E-05 0.029314834

2.56093E-05 0.041156595

3.97118E-05 0.056979252

6.09553E-05 0.07776421

9.26132E-05 0.104584692

0.000139285 0.138546642

0.000207351 0.180695003

0.000305546 0.231879187

0.000445674 0.292574045

0.000643469 0.36265682

0.000919619 0.441146806

0.001300942 0.525922553

0.001821704 0.613441086

0.002525035 0.698494025

0.003464389 0.774045163

0.004704958 0.831201439

0.006324913 0.859372041

0.008416337 0.846666498

0.011085658 0.780570136

0.014453386 0.648912863

0.018652949 0.441115078

0.023828414 0.149653847

0.030130931 -0.22835296

0.03771375 -0.689592056

0.046725789 -1.223215123

0.057303789 -1.809922867

0.069563239 -2.421682456

0.083588399 -3.022261044

0.099421884 -3.568700224

0.117054396 -4.013768631

0.136415346 -4.309318783

0.157365126 -4.410350549

0.179689839 -4.279462529

  

2 4

2

336

24

z

    z z 

 

 



 

1 3

 2

( ) 1 3 

6 24

z z

G

f z z z 

 

 

    

 

( )z
0.203099244 -3.891271841

0.227228529 -3.236319862

0.251644341 -2.323971263

0.275855243 -1.183865354

0.299326439 0.134406501

0.321498297 1.563553778

0.341807853 3.022718637

0.359712192 4.42321406

0.37471238 5.675389821

0.386376486 6.69598638

0.394360216 7.415244478

0.398423793 7.783031387

0.398443914 7.773328313

0.394419966 7.386589552

0.386474058 6.649713024

0.374844865 5.613624598

0.359875719 4.348741234

0.34199778 2.938806238

0.32170943 1.473754994

0.299553264 0.042351745

0.276092166 -1.274670276

0.25188591 -2.410326736

0.227469632 -3.315482119

0.203335281 -3.961062377

0.17991684 -4.33833279

0.157579839 -4.457401349

0.136615273 -4.344277538

0.117237788 -4.036930511

0.099587708 -3.580841496

0.083736272 -3.024534073

0.069693339 -2.415502093

0.05741676 -1.796853298

0.046822635 -1.20486494

0.037795734 -0.667523509

0.030199481 -0.204009302

0.023885038 0.175001038

0.018699163 0.466398593

0.014490659 0.673284083

0.011115369 0.80339742

0.008439746 0.867521063

0.006343145 0.878004905

0.004718997 0.847514951

0.003475077 0.788062225

0.002533081 0.710327622

0.001827693 0.623266057

0.001305351 0.533950936

0.000922829 0.447607457
0.00064578 0.36777953

0.00044732 0.296577985

0.000306705 0.234965214

0.000208159 0.183041252

0.000139842 0.140306712

9.29928E-05 0.105887771

6.12113E-05 0.078716542

3.98826E-05 0.057666418

2.5722E-05 0.041646221

1.64209E-05 0.029659389

1.03767E-05 0.020835511

6.49066E-06 0.014441379

4.01874E-06 0.009878044

2.46298E-06 0.006669237

1.49417E-06 0.004445294
Absolute values

0.004395759 0.000221986

0.006594452 0.00066604

0.009766478 0.000986414

0.014276926 0.00144197

0.020596014 0.002080197

0.029314834 0.002960798

0.041156595 0.004156816

0.056979252 0.005754904

0.07776421 0.007854185

0.104584692 0.010563054

0.138546642 0.013993211

0.180695003 0.018250195

0.231879187 0.023419798

0.292574045 0.029549979

0.36265682 0.036628339

0.441146806 0.044555827

0.525922553 0.053118178

0.613441086 0.06195755

0.698494025 0.070547897

0.774045163 0.078178561

0.831201439 0.083951345

0.859372041 0.086796576

0.846666498 0.085513316

0.780570136 0.078837584

0.648912863 0.065540199

0.441115078 0.044552623

0.149653847 0.015115039

0.22835296 0.023063649

0.689592056 0.069648798

1.223215123 0.123544727

1.809922867 0.18280221

2.421682456 0.244589928

3.022261044 0.305248365

3.568700224 0.360438723

4.013768631 0.405390632

4.309318783 0.435241197

4.410350549 0.445445405

4.279462529 0.432225715

 

2 4

2

336

( )

24

z

z z

z z





   

  



1

( ) 2 ( ) ( )

2

M

i

i

x

f a f x f b



  

      
3.891271841 0.393018456

3.236319862 0.326868306

2.323971263 0.234721098

1.183865354 0.119570401

0.134406501 0.013575057

1.563553778 0.157918932

3.022718637 0.305294582

4.42321406 0.44674462

5.675389821 0.573214372

6.69598638 0.676294624

7.415244478 0.748939692

7.783031387 0.78608617

7.773328313 0.78510616

7.386589552 0.746045545

6.649713024 0.671621015

5.613624598 0.566976084

4.348741234 0.439222865

2.938806238 0.29681943

1.473754994 0.148849254

0.042351745 0.004277526

1.274670276 0.128741698

2.410326736 0.243443

3.315482119 0.334863694

3.961062377 0.4000673

4.33833279 0.438171612

4.457401349 0.450197536

4.344277538 0.438772031

4.036930511 0.407729982

3.580841496 0.361664991

3.024534073 0.305477941

2.415502093 0.243965711

1.796853298 0.181482183

1.20486494 0.121691359

0.667523509 0.067419874

0.204009302 0.02060494

0.175001038 0.017675105

0.466398593 0.047106258

0.673284083 0.068001692

0.80339742 0.081143139

0.867521063 0.087619627

0.878004905 0.088678495

0.847514951 0.08559901

0.788062225 0.079594285

0.710327622 0.07174309

0.623266057 0.062949872

0.533950936 0.053929044

0.447607457 0.045208353
0.36777953 0.037145733

0.296577985 0.029954376

0.234965214 0.023731487

0.183041252 0.018487166

0.140306712 0.014170978

0.105887771 0.010694665

0.078716542 0.007950371

0.057666418 0.005824308

0.041646221 0.004206268

0.029659389 0.002995598

0.020835511 0.002104387

0.014441379 0.001458579

0.009878044 0.000997682

0.006669237 0.000673593

0.004445294 0.000224487
Cumulative normalization

0.000221986 1.2692E-05 9

0.000888025 5.07729E-05 10

0.00187444 0.000107171 11

0.003316409 0.000189616 12

0.005396607 0.000308551 13

0.008357405 0.000477835 14

0.012514221 0.000715501 15

0.018269125 0.001044537 16

0.026123311 0.001493601 17

0.036686365 0.002097543 18

0.050679575 0.002897605 19

0.068929771 0.003941061 20

0.092349569 0.005280088 21

0.121899547 0.006969608 22

0.158527886 0.009063834 23

0.203083713 0.011611314 24

0.256201891 0.014648347 25

0.318159441 0.01819077 26

0.388707337 0.022224347 27

0.466885899 0.026694208 28

0.550837244 0.031494127 29

0.63763382 0.036456722 30

0.723147137 0.041345947 31

0.80198472 0.045853487 32

0.86752492 0.049600749 33

0.912077543 0.052148046 34

0.927192581 0.053012248 35

0.95025623 0.054330913 36

1.019905028 0.058313084 37

1.143449755 0.065376755 38

1.326251965 0.075828474 39

1.570841893 0.089812907 40

1.876090258 0.107265486 41

2.236528981 0.127873575 42

2.641919613 0.15105179 43

3.07716081 0.175936711 44

3.522606215 0.201405058 45

3.95483193 0.226117569 46

( f x ) ( f b)



 




4.347850386 0.248588405 47

4.674718692 0.267277105 48

4.90943979 0.280697287 49

5.029010191 0.287533726 50

5.042585247 0.28830988 51

5.200504179 0.297338897 52

5.505798761 0.314794119 53

5.952543381 0.340336749 54

6.525757753 0.373110288 55

7.202052377 0.411777443 56

7.95099207 0.45459808 57

8.73707824 0.499542567 58

9.522184399 0.544431023 59

10.26822994 0.587086187 60

10.93985096 0.625486127 61

11.50682704 0.657902993 62

11.94604991 0.683015566 63

12.24286934 0.699986221 64

12.39171859 0.708496679 65

12.39599612 0.708741247 66

12.52473782 0.716102055 67

12.76818082 0.730020912 68

13.10304451 0.749166748 69

13.50311181 0.772040602 70

13.94128342 0.797093069 71

14.39148096 0.82283312 72

14.83025299 0.847919916 73

15.23798297 0.871231884 74

15.59964796 0.891910085 75

15.9051259 0.90937579 76

16.14909162 0.923324533 77

16.3305738 0.933700779 78

16.45226516 0.940658484 79

16.51968503 0.944513216 80

16.54028997 0.945691304 81

16.55796508 0.946701879 82

16.60507133 0.949395179 83

16.67307303 0.953283177 84

16.75421617 0.957922537 85

16.84183579 0.962932191 86

16.93051429 0.968002385 87

17.0161133 0.972896509 88

17.09570758 0.977447314 89

17.16745067 0.981549226 90

17.23040055 0.985148386 91

17.28432959 0.988231779 92

17.32953794 0.990816568x 93
17.36668368 0.992940375 x 94

17.39663805 0.994653017 x 95

17.42036954 0.996009865 x 96

17.43885671 0.997066869 x 97

17.45302768 0.997877095 x 98

17.46372235 0.998488563 x 99

17.47167272 0.998943125 x 100

17.47749703 0.99927613 x 101

17.4817033 0.999516624 x 102

17.48469889 0.999687897 x 103

17.48680328 0.999808215 x 104

17.48826186 0.99989161 x 105

17.48925954 0.999948652 x 106

17.48993314 0.999987165 x 107

17.49015762 1 x 108

882.453626
The UPL analysis for non-normal data distributions is included because inclusion of an outlier results in a non-normal data dis
r results in a non-normal data distribution.
Step 1. Identify size of sample set.

Large: When N (total number of sources) ≥ 20. For a large sample set can assume normal distribution based on 

the Central Limit Theorem; use the "UPL Pooled Template " .

Small: When N (total number of sources) < 20. For a small sample set the Central Limit Theorem cannot be used 

to support the normality assumption; move on to Step 2.

Step 2. Identify Normality of sample set.

Statistical tests of the kurtosis and skewness are used to evaluate the normality assumption.

If you have three or less runs in your entire sample set, skip down to Step 6. 

If you have more than three runs in your entire sample set, start in the "UPL Pooled Template " and test for 

normality in cells B45:B51. If the Kurtosis and Skewness tests show that raw data is Normal, you may continue to 

If the Kurtosis and Skewness tests show that raw data is Not Normal, move on to Step 3.

Step 3. Transform raw data to lognormal data.

In the "Lognormal Template ", take the LN() of the raw data in cells B55:L89 and test the normality again of the LNdata transformed in cells C45:C51.

If Kurtosis and Skewness tests show that the LN-data transformed is Lognormal, continue to use the "Lognormal 

If Kurtosis and Skewness tests show that the LN-data transformed is Not Lognormal, move on to Step 4.

Step 4. Apply Skewness Adjustment.

For data that is Not Normal and Not Lognormal, use the "Skewness Adjustment Template ". This is essentially

the UPL Pooled Template with a Skewness Adjustment. Note: DO NOT not use this template if n<13 (n is number 

For data that is Not Normal and Not Lognormal with n<13, move on to Step 5.

Step 5. 

See "Step 5. Intructions " tab

Step 6. 

See "Step 6. Intructions " tab

Things to keep in mind:

Type of data affects m (number of future runs) that will be used in calculation, for example:

If the average of three test runs is used to determine compliance; m=3 

If 30-day averages are used to determine compliance; m=30 

N = total number of sources

n = number of individual runs

df = degrees of freedom = n - 1

X = mean of data

s

2

 = pooled vaiance of data

m = number of future runs

tdf = t-statistic: t-distribution with df degrees of freedom at .99 confidence level 
Step 5. Instructions

For data sets that are Non Normal & Non Lognormal with n<13

1) Go to the "Template_nonnormal " tab. The adjustment starts in Row 150. Copy the kurtosis value (B41) and 

skewness value (B44) in the "Template_nonnormal " tab into cells H14 and H12 respectively of the "Calculations 

for Template " tab.

2) Cell E114 of the "Template_nonnormal " tab shows the actual confidence level (1 - 'Calculations for 

Template'!J139) after accounting for the skewness and kurtosis. If the confidence level is larger than 99.00% or 

lower than 99.00% then we need to adjust the t-statistic. 

3) Row 115 of the "Template_nonnormal " tab uses the IF function in EXCEL to conclude about the need to 

adjust or not. If you don’t need to adjust (because the difference between the actual level and 99.00% is not that 

large) then you can use the t-statistic in cell H102 of the "Template_nonnormal " tab. If the difference is 

significant, then you need to adjust the t-statistic.

4) The “Calculations_for_Template ” tab has the calculations for the confidence level for the first t-statistic in cell 

H102 from the“Template_nonnormal ” tab. Input values for this tab are obtained from the 

“Template_nonnormal ” tab, so you don’t need to worry about it.

5) To adjust the t-statistic go to the “Recalculate t-stat ” tab. This tab starts with the t-statistic in cell H102 from 

the “Template_nonnormal ” tab. If the confidence level from cell E114 of the "Template_nonnormal " tab was 

larger or lower THAN 99.00% you need to decrease or increase the t-statistic so you get closer to 99.00%. Cell G3 

has the value of the increase also called "interval", the candidate t-values will be calculated as the t-value in cell 

H102 plus multiples of the interval, for example if the t-value is 2, and the interval is 0.3, then the candidate t-values 

are: 2.03, 2.06, 2.09, 3.12, etc. These are calculated automatically for you in row 5 after a value is placed in cell 

G3. The corresponding p-values and confidence levels are also calculated automatically for you. Cell G3, the 

“interval” can be manipulated. For example, start with 0.1 and notice what the values in row 9 are. If they are not 

99.00% keep increaseing or decreasing the value in G3 until one of the values in row 9 is 99.00% and you get a 

"YES" value in row 11 in the same column. You may have to go out to several significant figures.

Calculations are in the tabs: Recalculations1 to Recalculations6. You don’t need to worry about them, they are 

basically duplicates of tab “Calculations_for_Template” and each one use a different value of the candidate tvalues in row 5. 

6) Sometimes the actual significance level in cell E114 of tab “Template_nonnormal ” is larger than 99%, then 

you have to use a negative interval (meaning that your value in cell G3, tab “Recalculate t-stat ” has to be 

negative), for example instead of 0.389, it has to be -0.389 or another negative number.

7) Once you get the adjusted t-value from tab “Recalculate t-stat ”, you copy the t-value and paste special the 

value into cell E119 in tab “Template_nonnormal ”.

8) The template will calculate the UPL for you in cell E122.
Runs

52071

Sandow

5B

1 7.0E-06

2 9.9E-06

3 1.5E-05

4 9.8E-06

5 1.3E-05

6 1.6E-05

7 5.8E-05 Outlier is included.

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

Emissions

26

27

28

29

30

31

32

33

34

35

Testing Normality Raw Data

Sample Size 7.00

Kurtosis 6.30

SE Kurtosis 1.00

Result Kurtosis non normal

Skewness 2.47

SE Skewness 0.79 RESULT

Result Skewness non normal non normal

### Make sure that kurtosis and skewness are copied in "Calculations for Template", in cells H14 and H12 respectively
ni = number test runs =

70000000000

Total Number of sources =1

n =Total # test runs =76

This step is for QC only:

means 1.85E-05

1.85E-05

Mean =



2

2

1

1

1

n

i

i

s xX

n



   

1 1

1

i

N

n

ij

i j

X x n    

1 1

n m

2 1 1

s

n m

    

 

1

1

N

i

i

dfn



 

      

2 1 1

s

n m

    

 

1

1

n

i

i ik

i

k

x x n   

1N

i

i

n

N

Pooled Variance = 3.20E-10

m= number future runs =3

Term1 0.4762

Term2 1.52E-10

Squared root Term2 1.23E-05  2 2 1 1 1 n i i

s xX

n



   

1 1

1

i

N

n

ij

i j

X x n    

1 1

n m

2 1 1

s

n m

    

 

1

1

N

i

i

dfn



 

      

2 1 1

s

n m

    

 

1

1

n

i

i ik

i

k

x x n   

1N

i

i

n

N
NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)

t-statistic = quantile t-distribution with df degrees of freedom at .99 confidence level = 3.142668

7

0.37792

99.13%

Conclusion

interval of differences for t-values used 0.453

5.860668403

Adjusted UPL for skewness and kurtosis for all sample size 9.08E-05 lb/MWh

9.08E-02 lb/GWhr

Confidence Level is not 99%, Go to recalculate tab to fix the Confidence Level

new t-statistic from tab Recalculate t-statistic

Adjustment for non-normality (skewness and kurtosis) for any sample size in particular for small samples 

a) n = number of test-runs =

b) Calculate u0

Current Confidence Level is

2

1

0

1

1

u

t

n







df t

2 1 1

s

n m

    

 

2

,

1 1 X tdf p

s

n m

 

     

9.08E-02 lb/GWhr

df t

2 1 1

s

n m

    

 

2

,

1 1 X tdf p

s

n m

 

     
Outlier is included.

4 and H12 respectively
Incomplete Beta Function approximation

where

Inputs skewness= 2.47

x=u0= 0.377920268 kurtosis= 6.30

n= 7

t0=t= 3.142668403

a= 3 a=

b= 0.5 b=

w= 0.303755 w=

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 ,

 

x

I a b 

0

1 1

,

2 2 u

n

I

  

   

w= 0.303755 w=

c.1.1. 0.003726024 d.1.1. 

c.1.2. 2 d.1.2. 

c.1.3 0.010342442 d.1.3

c.1.4. 1.69624451 d.1.4.

c.1.5. 2.666667 d.1.5.

c.1.6 2.531296 d.1.6

c.1.7 4.00E-01 d.1.7

Calculations Calculations

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( )

w a e w

a





1

2

a w

b

 
Result1: 2.48E-02 Result2:

1- Calculate 1.24E-02

2- Calculate

2.1. Calculate 0.025131

2.2. Calculate 7.809115

2.3. Calculate 49.02293

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

Result: 0.004003

3-Calculate

3.1. Calculate 0.301572

3.2. Calculate 0.326703

Result 6.09E-03

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     
4-Calculate

4.1. Calculate 6.20E-03

4.2. Calculate 6.13E-03

4.3. Calculate 7.39E-04

Result: 8.12E-04

5- Calculate

3.93E-02

5.1. Calculate

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          

5.1. Calculate

4.80E-02

5.2. Calculate

5.3. Calculate 9.74E-03

5.4. Calculate 3.25E-03

Result -2.24E-03

Corrected probability is::

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
4 a= 5

0.5 b= 0.5

0.303755 w= 0.303755

d.1.) Calculate e.1.) Calculate 



 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

0.303755 w= 0.303755

0.000278544 e.1.1. 1.6746E-05

6 e.1.2. 24

0.001047191 e.1.3 7.95225E-05

2.69624451 e.1.4. 3.69624451

11 e.1.5.

5.163843 e.1.6

5.38E-01 e.1.7

Calculations Calculations

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
9.53E-03 Result3:

 

0

3 4 1 1

,

24 2 2

u

n 

n

I

n

     

  

 

 

0

3 4 1 1

,

24 2 2

u

n 

n

I

n

     

  

 
 

0

3 4 1 1

,

24 2 2

u

n 

n

I

n

     

  

 

1 1 

,

2 2

5 1

,

2 2

n 



 



 







0.00868

 

0

3 4 1 1

,

24 2 2

u

n 

n

I

n

     

  

 

1 1 

,

2 2

5 1

,

2 2

n 



 



 






a= 5

b= 0.5

w= 0.303755

0 f.1.) Calculate 

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

w= 0.303755

f.1.1. 1.6746E-05

f.1.2. 24

f.1.3 7.95225E-05

f.1.4. 3.69624451

28 f.1.5. 28

8.707657 f.1.6 8.707657

6.77E-01 f.1.7 6.77E-01

Calculations

0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
1.88E-03 Result4: 1.88E-03

a= 3 a= 4

b= 1 b= 1

w= 0.607511 w= 0.607511

g.1.) Calculate 0 h.1.) Calculate 

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

w= 0.607511 w= 0.607511

g.1.1. 0.023864 h.1.1. 

g.1.2. 2 h.1.2. 

g.1.3 0.061065 h.1.3

g.1.4. 0.696245 h.1.4.

g.1.5. 2.666667 h.1.5.

g.1.6 5.062592 h.1.6

g.1.7 1.60E+00 h.1.7

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
Result5: 4.06E-02 Result6:

0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0.003509

6

0.012366

1.196245

11

10.32769

2.15E+00

0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
2.38E-02
3.142668

0.453

3.595668 4.048668 4.501668 4.954668 5.407668 5.860668

0.043638 0.03994 0.029781 0.020908 0.014444 0.009991

95.64% 96.01% 97.02% 97.91% 98.56% 99.00%

NO NO NO NO NO YES

4) If you don't get a YES in row 11, play around with the cell G3, the interval for differences until you get one p-value closed to 0.01

5) Once you get a YES, use the t-statistic in the calculations of the UPL.

Go back to Template_nonnormal tab and paste the value in row 5 that has a Yes in row 11.

Copy the new t-value in cell E119 Template_nonnormal

1) Start with initial t-statistic from Template

2) Determine interval of differences for t-values

2) Candidate values for t-statistic

3) Corrected p-values

Acutal significance level with t-value

4) Is p-value closed enough to 0.01?
t-statistic=

Incomplete Beta Function approximation

where

Inputs skewness= 2.47

x=u0= 0.316976781 kurtosis= 6.30

n= 7

t0=t= 3.142668403

a= 3 a=

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

2

1

0

1

1

u

t

n







a= 3 a=

b= 0.5 b=

w= 0.23204 w=

c.1.1. 0.001751454 d.1.1. 

c.1.2. 2 d.1.2. 

c.1.3 0.004953161 d.1.3

c.1.4. 1.767960465 d.1.4.

c.1.5. 2.666667 d.1.5.

c.1.6 1.933663 d.1.6

Calculations Calculations

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 2.33E-01 d.1.7

Result1: 1.44E-02 Result2:

1- Calculate 7.19E-03

2- Calculate

2.1. Calculate 0.025131

2.2. Calculate 7.809115

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



2.3. Calculate 49.02293

Result: 0.004003

3-Calculate

3.1. Calculate 0.301572

3.2. Calculate 0.326703

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result 5.49E-03

4-Calculate

4.1. Calculate 3.60E-03

4.2. Calculate 2.51E-03

4.3. Calculate 2.23E-04

Result: 1.31E-03

5- Calculate

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

2.28E-02

5.1. Calculate

1.96E-02

5.2. Calculate

5.3. Calculate 2.94E-03

5.4. Calculate 9.79E-04

Result 5.11E-03

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
3.595668 u0= 0.316977

4 a= 5

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

4 a= 5

0.5 b= 0.5

0.23204 w= 0.23204

0.0001004 e.1.1. 4.62259E-06

6 e.1.2. 24

0.00038311 e.1.3 2.22242E-05

2.767960465 e.1.4. 3.767960465

11 e.1.5.

3.944672 e.1.6

Calculations

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
3.14E-01 e.1.7

3.90E-03 Result3:

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
 

0

3 

1 1

,

4 2 2

u

n 

n

I

n

 

   

  

 

 

0

3 

1 1

,

4 2 2

u

n 

n

I

n

 

   

  

 
0.04364
a= 5

0 f.1.) Calculate 

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

a= 5

b= 0.5

w= 0.23204

f.1.1. 4.62259E-06

f.1.2. 24

f.1.3 2.22242E-05

f.1.4. 3.767960465

28 f.1.5. 28

6.6518 f.1.6 6.6518

Calculations

0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
3.95E-01 f.1.7 3.95E-01

5.67E-04 Result4: 5.67E-04

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 


a= 3 a= 4

g.1.) Calculate 0 h.1.) Calculate 

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

a= 3 a= 4

b= 1 b= 1

w= 0.464079 w= 0.464079

g.1.1. 0.01181 h.1.1. 

g.1.2. 2 h.1.2. 

g.1.3 0.031419 h.1.3

g.1.4. 0.76796 h.1.4.

g.1.5. 2.666667 h.1.5.

g.1.6 3.867326 h.1.6

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 9.33E-01 h.1.7

Result5: 2.75E-02 Result6:

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 


0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

0.001337

6

0.00486

1.26796

11

7.889344

0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
1.26E+00

1.16E-02

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs skewness= 2.47

x=u0= 0.267956241 kurtosis= 6.30

n= 7

t0=t= 3.142668403

a= 3 a=

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

2

1

0

1

1

u

t

n







a= 3 a=

b= 0.5 b=

w= 0.183019 w=

c.1.1. 0.000891254 d.1.1. 

c.1.2. 2 d.1.2. 

c.1.3 0.002552562 d.1.3

c.1.4. 1.816980722 d.1.4.

c.1.5. 2.666667 d.1.5.

c.1.6 1.525161 d.1.6

Calculations Calculations

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 1.45E-01 d.1.7

Result1: 8.36E-03 Result2:

1- Calculate 4.18E-03

2- Calculate

2.1. Calculate 0.025131

2.2. Calculate 7.809115

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



2.3. Calculate 49.02293

Result: 0.004003

3-Calculate

3.1. Calculate 0.301572

3.2. Calculate 0.326703

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result 4.08E-03

4-Calculate

4.1. Calculate 2.09E-03

4.2. Calculate 1.10E-03

4.3. Calculate 7.51E-05

Result: 1.07E-03

5- Calculate

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

1.32E-02

5.1. Calculate

8.58E-03

5.2. Calculate

5.3. Calculate 9.90E-04

5.4. Calculate 3.30E-04

Result 5.32E-03

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
4.048668 u0= 0.267956

4 a= 5

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

4 a= 5

0.5 b= 0.5

0.183019 w= 0.183019

4.04003E-05 e.1.1. 1.46964E-06

6 e.1.2. 24

0.000155723 e.1.3 7.12506E-06

2.816980722 e.1.4. 3.816980722

11 e.1.5.

3.111328 e.1.6

Calculations

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
1.95E-01 e.1.7

1.70E-03 Result3:

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
 

0

3 

1 1

,

4 2 2

u

n 

n

I

n

 

   

  

 

 

0

3 

1 1

,

4 2 2

u

n 

n

I

n

 

   

  

 
0.03994
a= 5

0 f.1.) Calculate 

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

a= 5

b= 0.5

w= 0.183019

f.1.1. 1.46964E-06

f.1.2. 24

f.1.3 7.12506E-06

f.1.4. 3.816980722

28 f.1.5. 28

5.246553 f.1.6 5.246553

Calculations

0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
2.46E-01 f.1.7 2.46E-01

1.91E-04 Result4: 1.91E-04

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 


a= 3 a= 4

g.1.) Calculate 0 h.1.) Calculate 

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

a= 3 a= 4

b= 1 b= 1

w= 0.366039 w= 0.366039

g.1.1. 0.006228 h.1.1. 

g.1.2. 2 h.1.2. 

g.1.3 0.017005 h.1.3

g.1.4. 0.816981 h.1.4.

g.1.5. 2.666667 h.1.5.

g.1.6 3.050321 h.1.6

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 5.81E-01 h.1.7

Result5: 1.77E-02 Result6:

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 


0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

0.000559

6

0.002075

1.316981

11

6.222655

0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
7.82E-01

5.70E-03

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs skewness= 2.47

x=u0= 0.228440731 kurtosis= 6.30

n= 7

t0=t= 3.142668403

a= 3 a=

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

2

1

0

1

1

u

t

n







a= 3 a=

b= 0.5 b=

w= 0.148038 w=

c.1.1. 0.000484097 d.1.1. 

c.1.2. 2 d.1.2. 

c.1.3 0.001398945 d.1.3

c.1.4. 1.851961645 d.1.4.

c.1.5. 2.666667 d.1.5.

c.1.6 1.233653 d.1.6

Calculations Calculations

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 9.50E-02 d.1.7

Result1: 4.97E-03 Result2:

1- Calculate 2.48E-03

2- Calculate

2.1. Calculate 0.025131

2.2. Calculate 7.809115

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



2.3. Calculate 49.02293

Result: 0.004003

3-Calculate

3.1. Calculate 0.301572

3.2. Calculate 0.326703

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result 2.83E-03

4-Calculate

4.1. Calculate 1.24E-03

4.2. Calculate 5.11E-04

4.3. Calculate 2.79E-05

Result: 7.59E-04

5- Calculate

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

7.87E-03

5.1. Calculate

4.00E-03

5.2. Calculate

5.3. Calculate 3.67E-04

5.4. Calculate 1.22E-04

Result 4.11E-03

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
4.501668 u0= 0.228441

4 a= 5

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

4 a= 5

0.5 b= 0.5

0.148038 w= 0.148038

1.7782E-05 e.1.1. 5.23852E-07

6 e.1.2. 24

6.90325E-05 e.1.3 2.55486E-06

2.851961645 e.1.4. 3.851961645

11 e.1.5.

2.516652 e.1.6

Calculations

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
1.28E-01 e.1.7

7.94E-04 Result3:

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
 

0

3 

1 1

,

4 2 2

u

n 

n

I

n

 

   

  

 

 

0

3 

1 1

,

4 2 2

u

n 

n

I

n

 

   

  

 
0.02978
a= 5

0 f.1.) Calculate 

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

a= 5

b= 0.5

w= 0.148038

f.1.1. 5.23852E-07

f.1.2. 24

f.1.3 2.55486E-06

f.1.4. 3.851961645

28 f.1.5. 28

4.243766 f.1.6 4.243766

Calculations

0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
1.61E-01 f.1.7 1.61E-01

7.10E-05 Result4: 7.10E-05

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 


a= 3 a= 4

g.1.) Calculate 0 h.1.) Calculate 

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

a= 3 a= 4

b= 1 b= 1

w= 0.296077 w= 0.296077

g.1.1. 0.00347 h.1.1. 

g.1.2. 2 h.1.2. 

g.1.3 0.009652 h.1.3

g.1.4. 0.851962 h.1.4.

g.1.5. 2.666667 h.1.5.

g.1.6 2.467306 h.1.6

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 3.80E-01 h.1.7

Result5: 1.13E-02 Result6:

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 


0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

0.000253

6

0.000953

1.351962

11

5.033304

0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
5.11E-01

2.87E-03

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs skewness= 2.47

x=u0= 0.196407452 kurtosis= 6.30

n= 7

t0=t= 3.142668403

a= 3 a=

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

2

1

0

1

1

u

t

n







a= 3 a=

b= 0.5 b=

w= 0.122206 w=

c.1.1. 0.000277615 d.1.1. 

c.1.2. 2 d.1.2. 

c.1.3 0.000807556 d.1.3

c.1.4. 1.87779413 d.1.4.

c.1.5. 2.666667 d.1.5.

c.1.6 1.018382 d.1.6

Calculations Calculations

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 6.47E-02 d.1.7

Result1: 3.04E-03 Result2:

1- Calculate 1.52E-03

2- Calculate

2.1. Calculate 0.025131

2.2. Calculate 7.809115

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



2.3. Calculate 49.02293

Result: 0.004003

3-Calculate

3.1. Calculate 0.301572

3.2. Calculate 0.326703

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result 1.93E-03

4-Calculate

4.1. Calculate 7.59E-04

4.2. Calculate 2.52E-04

4.3. Calculate 1.12E-05

Result: 5.18E-04

5- Calculate

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

4.81E-03

5.1. Calculate

1.98E-03

5.2. Calculate

5.3. Calculate 1.48E-04

5.4. Calculate 4.94E-05

Result 2.93E-03

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
4.954668 u0= 0.196407

4 a= 5

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

4 a= 5

0.5 b= 0.5

0.122206 w= 0.122206

8.42918E-06 e.1.1. 2.05171E-07

6 e.1.2. 24

3.2896E-05 e.1.3 1.00502E-06

2.87779413 e.1.4. 3.87779413

11 e.1.5.

2.0775 e.1.6

Calculations

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
8.71E-02 e.1.7

3.92E-04 Result3:

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
 

0

3 

1 1

,

4 2 2

u

n 

n

I

n

 

   

  

 

 

0

3 

1 1

,

4 2 2

u

n 

n

I

n

 

   

  

 
0.02091
a= 5

0 f.1.) Calculate 

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

a= 5

b= 0.5

w= 0.122206

f.1.1. 2.05171E-07

f.1.2. 24

f.1.3 1.00502E-06

f.1.4. 3.87779413

28 f.1.5. 28

3.503235 f.1.6 3.503235

Calculations

0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
1.10E-01 f.1.7 1.10E-01

2.86E-05 Result4: 2.86E-05

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 


a= 3 a= 4

g.1.) Calculate 0 h.1.) Calculate 

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

a= 3 a= 4

b= 1 b= 1

w= 0.244412 w= 0.244412

g.1.1. 0.002028 h.1.1. 

g.1.2. 2 h.1.2. 

g.1.3 0.005717 h.1.3

g.1.4. 0.877794 h.1.4.

g.1.5. 2.666667 h.1.5.

g.1.6 2.036764 h.1.6

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 2.59E-01 h.1.7

Result5: 7.29E-03 Result6:

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 


0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

0.000122

6

0.000466

1.377794

11

4.155

0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
3.48E-01

1.50E-03

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs skewness= 2.47

x=u0= 0.170247165 kurtosis= 6.30

n= 7

t0=t= 3.142668403

a= 3 a=

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

2

1

0

1

1

u

t

n







a= 3 a=

b= 0.5 b=

w= 0.102589 w=

c.1.1. 0.000166657 d.1.1. 

c.1.2. 2 d.1.2. 

c.1.3 0.000487214 d.1.3

c.1.4. 1.897410917 d.1.4.

c.1.5. 2.666667 d.1.5.

c.1.6 0.854909 d.1.6

Calculations Calculations

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 4.56E-02 d.1.7

Result1: 1.91E-03 Result2:

1- Calculate 9.56E-04

2- Calculate

2.1. Calculate 0.025131

2.2. Calculate 7.809115

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



2.3. Calculate 49.02293

Result: 0.004003

3-Calculate

3.1. Calculate 0.301572

3.2. Calculate 0.326703

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result 1.31E-03

4-Calculate

4.1. Calculate 4.78E-04

4.2. Calculate 1.31E-04

4.3. Calculate 4.88E-06

Result: 3.52E-04

5- Calculate

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

3.03E-03

5.1. Calculate

1.03E-03

5.2. Calculate

5.3. Calculate 6.43E-05

5.4. Calculate 2.14E-05

Result 2.04E-03

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
5.407668 u0= 0.170247

4 a= 5

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

4 a= 5

0.5 b= 0.5

0.102589 w= 0.102589

4.25218E-06 e.1.1. 8.69445E-08

6 e.1.2. 24

1.66609E-05 e.1.3 4.27308E-07

2.897410917 e.1.4. 3.897410917

11 e.1.5.

1.744014 e.1.6

Calculations

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
6.14E-02 e.1.7

2.04E-04 Result3:

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
 

0

3 

1 1

,

4 2 2

u

n 

n

I

n

 

   

  

 

 

0

3 

1 1

,

4 2 2

u

n 

n

I

n

 

   

  

 
0.01444
a= 5

0 f.1.) Calculate 

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

a= 5

b= 0.5

w= 0.102589

f.1.1. 8.69445E-08

f.1.2. 24

f.1.3 4.27308E-07

f.1.4. 3.897410917

28 f.1.5. 28

2.940887 f.1.6 2.940887

Calculations

0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
7.72E-02 f.1.7 7.72E-02

1.24E-05 Result4: 1.24E-05

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 


a= 3 a= 4

g.1.) Calculate 0 h.1.) Calculate 

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

a= 3 a= 4

b= 1 b= 1

w= 0.205178 w= 0.205178

g.1.1. 0.001235 h.1.1. 

g.1.2. 2 h.1.2. 

g.1.3 0.003518 h.1.3

g.1.4. 0.897411 h.1.4.

g.1.5. 2.666667 h.1.5.

g.1.6 1.709818 h.1.6

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 1.82E-01 h.1.7

Result5: 4.77E-03 Result6:

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 


0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

6.27E-05

6

0.000241

1.397411

11

3.488029

0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
2.46E-01

8.13E-04

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
t-statistic=

Incomplete Beta Function approximation

where

Inputs skewness= 2.47

x=u0= 0.148708341 kurtosis= 6.30

n= 7

t0=t= 3.142668403

a= 3 a=

Calculate u0. Sustitute each candidate t-value in u0 equation

Calculate the Incomplete Beta Functions needed for adjustment

c.1.) Calculate d.1.) Calculate 

1

x

w b

x

 

     

 , 

 

x

I a b   

0

1 1

,

2 2 u

n

I

  

   

2

1

0

1

1

u

t

n







a= 3 a=

b= 0.5 b=

w= 0.087343 w=

c.1.1. 0.000104026 d.1.1. 

c.1.2. 2 d.1.2. 

c.1.3 0.000305294 d.1.3

c.1.4. 1.912657231 d.1.4.

c.1.5. 2.666667 d.1.5.

c.1.6 0.727856 d.1.6

Calculations Calculations

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

2

1

0

1

1

u

t

n






c.1.7 3.31E-02 d.1.7

Result1: 1.24E-03 Result2:

1- Calculate 6.19E-04

2- Calculate

2.1. Calculate 0.025131

2.2. Calculate 7.809115

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n



2.3. Calculate 49.02293

Result: 0.004003

3-Calculate

3.1. Calculate 0.301572

3.2. Calculate 0.326703

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

0 0

1 1 1

( ) ,

2 2 2

u

n

P t I

  

    

3

0 0

2 1 1 1 1

( ) ,1 ,1

6 2 2 2 3 2 u u

n n n n P t I I

n n



 

       

         

2

2

0

1

2 2

0

2 1 1

1 1

( )

6 2

1

1

n

n

t

n

P t

n

t

n







  



    



 

  

  

2 1

6 2

n

n



2

0

2 1 1

1

n

t

n







1

2 2

0 1

1

n

t

n



 

     

1

3 2

n

n



2 1

6 2

n

n


Result 9.02E-04

4-Calculate

4.1. Calculate 3.09E-04

4.2. Calculate 7.15E-05

4.3. Calculate 2.25E-06

Result: 2.40E-04

5- Calculate

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   

1.96E-03

5.1. Calculate

5.60E-04

5.2. Calculate

5.3. Calculate 2.96E-05

5.4. Calculate 9.87E-06

Result 1.42E-03

     

4 0 0 0

1 1 1 1 1 2 1 3 4 1 1

( ) , , ,

24 2 2 12 2 2 24 2 2 u u u

n n n n n n n n

P t I I I

n n



             

        

     

0

1 1 1

,

24 2 2 u

n n I       

  

0

1 2 1 1

,

12 2 2 u

n n n

I

n

    

   

  

0

4 1 3 1

,

24 2 2 u

n n n

I

n

    

   

    

     

2

3/2 0 0

2 2

0 0

1 (2 5) 1 1 1 2 5 8 1 1 ( ) , ,

72 2 2 24 2 2

1 2 5 12 1 3 1 2 5 12 5 1

, ,

24 2 2 72 2 2

u u

u u

n n n n n n n

P t I I

n

n n n n n n n n

I I

n n



          

      

   

                    

 

0

1 (2 5) 1 1

,

72 2 2 u

n n n

I

    

   

  

2

0

1 2 5 8 1 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 3 1

,

24 2 2 u

n n n n

I

n

     

   

  

2

0

1 2 5 12 5 1

,

72 2 2 u

n n n n

I

n

     

   
Corrected probability is::

3 4 3/2

2

0 3 P t( ) Skewness P t( ) Kurtosis P t( ) P t( )          
5.860668 u0= 0.148708

4 a= 5

e.1.) Calculate 

 

3

2 2

2

2 3

( , ) 1 1 5 3 1 3 11 1

,

( ) ( ) 2 2 2 3 2 326 3

3 1 1

 

2 6 2

w a a w e w a w a

I a b a a w a a

a a b b

w a w

       

                   

  

     

  

0

1 1

,

2 2 u

n

I

  

   

4 a= 5

0.5 b= 0.5

0.087343 w= 0.087343

2.26149E-06 e.1.1. 3.93891E-08

6 e.1.2. 24

8.88841E-06 e.1.3 1.94085E-07

2.912657231 e.1.4. 3.912657231

11 e.1.5.

1.484827 e.1.6

Calculations

0

1 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
4.45E-02 e.1.7

1.11E-04 Result3:

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
 

0

3 

1 1

,

4 2 2

u

n 

n

I

n

 

   

  

 

 

0

3 

1 1

,

4 2 2

u

n 

n

I

n

 

   

  

 
0.00999
a= 5

0 f.1.) Calculate 

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

a= 5

b= 0.5

w= 0.087343

f.1.1. 3.93891E-08

f.1.2. 24

f.1.3 1.94085E-07

f.1.4. 3.912657231

28 f.1.5. 28

2.503826 f.1.6 2.503826

Calculations

0

3 1

,

2 2 u

n

I

  

    0

5 1

,

2 2 u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
5.59E-02 f.1.7 5.59E-02

5.72E-06 Result4: 5.72E-06

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 


a= 3 a= 4

g.1.) Calculate 0 h.1.) Calculate 

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

a= 3 a= 4

b= 1 b= 1

w= 0.174686 w= 0.174686

g.1.1. 0.00078 h.1.1. 

g.1.2. 2 h.1.2. 

g.1.3 0.002238 h.1.3

g.1.4. 0.912657 h.1.4.

g.1.5. 2.666667 h.1.5.

g.1.6 1.455713 h.1.6

Calculations Calculations

0

1

,1

2

u

n

I

  

   

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

( , )

( ) a

( )

w a e w

a





1

2

a w

b

 

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
g.1.7 1.32E-01 h.1.7

Result5: 3.18E-03 Result6:

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 


0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     

3.38E-05

6

0.00013

1.412657

11

2.969654

0

1

,1

2

u

n

I

  

   

3

5 3 2 1

2 3 2 3

a

  a a 

3 1 2 1 1

2 6 3

w a a

 

     
1.78E-01

4.57E-04

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 

3 1 2 1 1

2 6 3

w a a

 

     

2 3 1

2 6

w a

    

 
Emissions (lb/MW) Hg

52071

ORIS/Facility/Unit Sandow

Runs 5B 1 7.2E-06 2 1.0E-05 3 1.0E-05 4 1.3E-05 5 1.7E-05 6 1.6E-05

Outlier Removed (This was a single 

run test because 2 of 3 runs failed 

spike recovery).==>

89

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

ni = number test Runs =6

Total Number of sources =1

1N

i

i

n



N
n =Total # test Runs = 6

5

This step is for QC only:

means 1.2E-05

1.2E-05

UPL Mean =

Pooled Variance = 1.4E-11

m= number future Runs = 3

Term1 0.5000

Term2 7.2E-12

Squared root Term2 2.7E-06

NOTE: the pvalue for the t-statistic is calculated as: 2*alpha, where 1-alpha is desired confidence, so if 99% confidence is desire

t-statistic

= quantile t-distribution with df degrees 

of freedom at .99 confidence level = 3.36

1

1

N

i

i

df n



 

      

d f t

1

i

i

n





 

2

2

1

1

1

n

i

i

s x X

n 

 





1 1

1

N ni

ij

i j

X x

n  

 

1 1

n m



s

2

1 1

n m

    

 

d f t

2

1 1

s

n m

    

 

1

1

ni

i ik

i k

x x

n 

 

  1 1
UPL POOLED VARIANCE = 2.13E-05

Normal UPL 2.02E-05

Lognormal UPL 2.15E-05

Final UPL (lognormal distribution) 2.15E-05

2

,

1 1 X tdf p

s

n m

 

     

e, so if 99% confidence is desired then alpha=0.01 and 2*alpha=2*(0.01)
lb/MW 2.13E-02 lb/GWhr

lb/MW

lb/MW 2.15E-02 lb/GWhr






2

,

1 1 X tdf p

s

n m

 

     
2

,

1 1 X tdf p

s

n m

 

     